DETERMINATION OF THE NONLINEARITY ON THE BASIS OF ITS DESCRffiING FUNCTION

DETERMINATION OF THE NONLINEARITY ON THE BASIS OF ITS DESCRffiING FUNCTION

By

J. SmILo

Department of .-\ntomation. Poly technical 1:niversity. Budapest (Received January 17. 1966)

Presented by Prof. F. CS . .\.KI

Introduction

The mathematical theory of optimal processes provides a possibility for the materialisation of control systems which realise the maximal poten- tialities of the systems. In the classical control theory this led to the synthesis of systems 'which possess the optimal dynamic properties. A means of realising such systems is the nonlinear correction. In spite of what was said above, in linear systems traditional frequency methods of correction have a decisive role. Methods based on these principlcs satisfy practical requirements and their realisation (as regards methods and means) is relatively simple.

Frequency methods are being applied also for the synthesis of nonlinear systems [5]. The result of the synthesis is a nonlinear correction of the system.

This trend of nonlinear correction may assume an important role in the future.

In conjunction with the problem of synthesis of nonlinear systems the task arises to find the inverse describing function, i.e. finding the nonlinear characteristic from its describing function.

An analytical approach to the solution of the problem of the inverse describing function was proposed by GIBSOl' and DITADA [1], but as the authors of this paper remark - the operations expressed by the equations

"are in general very difficult to perform in practical cases, if not impossible".

For this reason GIBSO::> and DITADA propose the use of numerical mcthods of solution.

In earlier works [2, 3, 4] a generalised method for determining the describing functions in the case of piecewise-linear, nonlinear systems 'with finite discontinuities, is described. This method permits direct solution of the problem of finding the inverse describing function. The solution is approx- imate.

The approximate character of the solution is duc to thc fact that the nonlinear characteristic is sought for (substituted) by a family of piecewise- linear nonlinear characteristics with finite discontinuities. Accordingly, the given describing function is quantized and the describing function of the non- linear characteristic to be determined will only coincide with the given values in selected points.

126 J. SOJILO

The possibility of applying the approximate method III conjunction 'with the given problem is also confirmed by the fact that the method of describing functions is approximate in itself.

It is pointed out that in his book GIBSO::> [5] mentions a method, the basic idea of 'which is to some extent similar to that presented in this paper.

1. Generalised method for determining the descrihing function

The describing function indicates how the base frequency of a periodic output signal 'with some nonlinearity relates to its input signal in the case of a harmonic input signal.

In the complex form the describing fUllction can be written as follows:

II b

ja

(1.1)

Let us assumc that the nonlinearitv IS characterized bv the following function:

y

=f(x)

(1.2)

where

y -

output signal of nonlinearity, x input signal of nonlinearity.

Let us assume that the input signal of nonlinearity 'will be:

x (1.3)

B sin (ut = B sin (f (1 A)

Xo constant component

x

harmonic component

B

amplitude

C') frequency

q: = C>Jt - phase angle

The coefficients of the describing function can be determined as follows:

b=

a.=

1

::rB

f(x)

sin

q dq

1 '

J ^f(:1.')

^{cos If}

^dq

::rB

(1.5)

(l.6)

DETERJIISATIO.Y OF THE .YOSLI.YEARI1T 127

The nonlinear characteristics at the input signal equ. (1.3) can bc con- sidered in a very general case [3, 4] as either having different branches for the phase quarters of the harmonic component (x = B sin (p), as is shown in Fig. 1.

Therefore the nonlinearity is charaeterised by the characteristics

11

(x),

12

^(x),

13

(x), flx) within the limits

o

- - : 7 : ^:T :7

2

- - : 7 :

3 2 .

3

:7 - 2:7 - respectively.

2

To obtain a similar method of investigation for all the phase quarters it is advisable to turn the branches

13

^(x) and

fl

(x) by 180~ with respect to the ongm of the coordinates (central projection). The so obtained branches

Fig. ]

of the characteristics will be denoted by

F or odd ;;ingle valucd characteristics, if xo 0:

j~ (x) I~ (x)

h'

(x) =

II'

(x) ^(1.7)

For odd two valued characteristics, if

Xo 0:

(1.8)

12

^{(x) =}

L

(x)

128 J. SOMLU

For odd single valued characteristics, if Xo / 0, and in the case of non- symmetrical single valued characteristics:

f1

(x) =

f2

(x)

f3'

(x) =

J1'

^(x)

(1.9)

In other cases (odd two valued characteristic, if ^Xo

+ °

and two valued non-symmetrical characteristic), the branches of the characteristic differ.

In the following only the piecewise linear characteristics with finite discontinuities will be considered. For the branches of the characteristics we

y ^...

>'

r,

r, Fi'f, Fn-f

x j= 1,2

B Fig. 2

will apply the designations as shown in Fig. 2 which is of a quite general type.

By suitable selection of the parameters

f(O);

Fj ; K j ; Dj; (i = 1, 2, 3, ... n) any shape of the branches of the characteristics can be realized.

The branch of the characteristic shown in Fig. 2 can be sub-divided into

"elementary" characteristics. To obtain generally applicable results we in- vestigated the characteristic jj (x) - (j

le

2, 3', 4.'). The "elementary"

characteristic will be:

fDo

(x) =

f(O)

P

^(x) =

{(I~i - I~j'd) X (K; K;+l)F;

fD{x) {O

D;

j"(x)

=

Knx

x>o

x<F;

x> ^F;

x<F;

x>F;

F

n - 1

< B

~ FIl

(1.10)

By means of the "elementary" characteristic it is possible to construct the branch of the nonlinear characteristic:

71-]

fj

(x)

= fD"

(x) ---L.

j"

(x) ~

[p

(x)

fD,

(x)]. (1.11)

;=1

DETERJILYATIOS OF THE _YOSLISEARITY 129

(For the purpose of simplicity it is not indicated in the righthand part of the formulae that the parameters relate to the j-th branch of the parameters.) In this way the branch of the nonlinear characteristic was substituted by a sum of simple elements which can be superimposed (these elements correspond to the relay characteristic, linear characteristic, linear charac- teristic with saturation and relay characteristics with dead zone).

Such a decomposition permits investigating any piecewise linear ^11011- linearity by a unified method.

By introducing a new variable it can easily be proved [3,

4,]

that the components of the coefficients of the describing function can be determined in a single way from the branches of the characteristics. Therefore, on the hasis of the formulae (1.5) and (1.6) we obtain: '

:r!~

b

= _1_

J' ^[f1

^(x)

~ f~

^(x)

+ f3'

(x)

+ f1'

(x)] sin q

dq;

;-cB

when'

and a where

b -_{i -}

:r/:2

(j = 1, :2, 3', 4')

1

j' ^fj

^{(x) sing dq}

;-cB •

__ 1_

f'Ji

^{(x) cos q dq}

7fB . (j = 1. 2, 3', 4')

(1.12)

(1.13)

(1.14)

(1.15)

By inserting equ. (1.11) into (1.13) and the integration taking into consider- ation equ. (1.10), we obtain the following generalised formula:

f(O) ,

1 {_ n--! [ _ _

B

b i = - -

B ~

- , ,1 K" -'-, . ' , - -

~'

K· -I K-'. ) k I , 1

1-) F

;-c '± ,=1 , ;.

-.0-11-~)'J-}'

^(1.16)

F[ ,F,

The function which figures in equ. (1.16) can be determined from the following expressions:

( B)

2 ( .

F;

_{11 1}

W)

k - - = - arc SIn - -I "

F;;-c B

B

(1.17)

zl-:; )

^{4 ;-c}

^B 11 ^{P B2 ,}

^(1.18)

130 ^{J. SOJIU5}

From the equ.

(1.12)

and

(1.16)

the coefficient "b" can be determined for a nonlinear characteristic of any shape. By inserting the function _{. ,} k

I-I

_Fi

B

_I ^and 1

I ~l

expressed by equ.

(1.17)

and

(1.18),

in the obtained expressions the

I

F_{i )}

coefficient

"b"

is obtained in its conventional form. The functions

k (~l

.Fii and l

(:i)

can be calculated in advance and tabulated. By means of these tables the numerical values of the coefficient "b" which are necessary for using graphical methods can easily be found. These tables are contained, for instance,

(

in an earlier work [2, 3] (it is pointed out that the function

k~)

B' ^{was used} for the first time by K. MAGl\US for describing functions).

In a similar manner the following generalised formula can be derived:

1 {2

f

(0)

a·= - - -

J 2:r B ^{J(.c.. )}

(2 _

^{Fj -)}

^-+-

/.1 B B~'

(1.19)

...L ") _ _ D /

i -

F.

B

/

B~

⁾

l}·

The coefficient "a" can be determined by means of the formulae

(1.14)

and

(1.19).

It IS pointed out that in a similar manner it is possible to obtain the generalised formulae for determining the constant of the component and the coefficients of the higher harmonics [4-].

In the case of symmetrical oscillations and single yalued characteristics:

b

_4f(0)

:rE

a

=

⁰

K,c..I)k(-_~-j·

"- F.

/ .

Fn)

In thc case of t,\-O valued characteristics:

2b~

The component b_j (j

= 1,2)

^IS determined from equ.

(1.16),

a = 2a[ - 2a~

The component aj (j

1,2)

is determined by means of equ.

(1.19).

( 1.20) (1.21)

(1.22)

(1.23)

DETER.1IISATIO.Y OF THE .YO.YLLYEARITY 131

Equs. (1.20)-(1.23) are correct eyen for such nonlinear characteristics where the shape depends on the amplitude or the frequency of the input signal.

In the following only the case of symmetrical oscillations will be in- yestigated. In practice this case is of decisive importance. The generalised for- mulae also allow the investigation of non-symmetrical oscillations.

In the following it is assumed that the shape of the nonlinear charac- teristics does not depend on the amplitude of the input signal.

H. Determination of the nonlinearity from the descrihing function 1. Single valued inverse characteristics

In the case of symmetrical oscillations the coefficients of the describing functions unequiyocally determine the appropriate nonlinear characteristic.

b

a8~---+~~~·~~---- 0,6

~:::::~~~--.--~---

..

""~---"."--~"~---"~--~----~.-~-~-::::::;;~~~~~~~

150 200 B

Fig. 3

These coefficients are functions of the amplitude

B

of the input signal of non- linearity.

If the coefficient

a

=

a(B)

= 0, then the nonlinear characteristic is single yalued.

The shape of the appropriate nonlinear characteristic depends on the form of the function b = b( B).

a)

Lct us assume that b b( B) is a continuous, single yalued function

·with continuous first deriYatiYes. In this cascf(O) =--=

Di

= 0; (i = 1,2 .. .

n-l)

(see Appendix 1).

On the basis of equ. (1.20) the inyerse nonlinear characteristic is deter- mined as follows:

Let us assume that the function b( B) has the form shown in Fig. 3 (in accordance with the condition of continuity of the first deriyatiyes of the functions in points where this condition is not fulfilled, ·we assume small rounding offs).

Let us separate the horizontal axis of Fig. 3 ,\"ith suitahle selected points.

This permits determining the hond points of the piecewise linear nonlinear characteristics.

132 J. SOJILU

According to formula (1.20) we consider a sub-series of the selected points one by one.

Let us assume that B = F_1• According to formula (1.20):

(2.1) For B F~

K2 (K1 - Kz) k I FF2l

=

b(Fz)

1 '

Therefrom:

(2.2)

K=O

100 150 200 x

Fig. -+

The -value of

Kl

is known from equ. (2.1) and therefore

K2

can be deter- mined by means of equ. (2.2).

By continuing the described procedure in this manner it is possible to determine the nonlinear characteristic in the rangc of interest to us. Thus, for

B F:

_{l :}

(2.3)

Generally, for

B

=

F,,:

b(FiJ) - ~ ^{(Ki - K i-}

¹⁾

^k I~~l ^K

^{h- i}

^k I~'-l

Kiz

= ~~~~----~--"""---~-"".=-'- -.~~----.~.-~ -'.:.--'.

1 - k l - - 1

: F;:_;

(2.4-)

The nonlinear characteristic is fully determined by the bend points (Fl'

F2 ... )

and the corresponding slope of the linear sections.

DETERJILYATIO,Y OF THE :YOSLL\EARln' 133

The describing function of the obtained nonlinear characteristic in the bend points

'will

coincide accurately with the values of the giyen describing function.

In principle the accuracy of the method can he increased to any degree hy increasing the numher of linear sections. In most practical cases even a small numher of sections giyes satisfactory results. If the accuracy loequire- ments are more stringent yery accurate results can he ohtained hy using digital computers. The given method is yery convenient for computer calcu- lations.

It is adyisahle to suh-diyide the horizontal axis of the graph of the function b(B) in accordance with the shape of the function. This means that in points where the changes in the descrihing function are large, it is advisahle to take the selected points near to each other. If the first derivative of the describing function is not a continuous function, then it is necessary to change over to characteristics with finite discontinuities to ohtain results 'whieh re- flect this discontinuity.

b)

If the nonlinear characteristic does not hegin in the origin of the coordinates (y 0; x = 0), i. e.

f(O)

0, then, as can he seen from equ. (1.20), b(B) ~::-G, if B -- 0

(

^{- . ; ) ') -)}

The values

f(O)

can he determined as follows. Inside the first linear section we select one further point (0

<

^F;

<

^Fj)' According to equ. (1.20):

(:2.6) b(F.J

From equ. (2.6) 'I-e can dctcrmine

f(O)

and

K

_{I •}

In those points for which the first dcrivative of thc describing function has an infinite discontinuity, the nOll-linear characteristic has a finite di5conti- nuity. Thc value of the finite discontinuity

Dr

can he determined in a manner similar to that of the value of

f(O).

Let us assume that the first derivative of the function b( B) has an in- finite discontinuity at the point B = Fr. \Vc select on the (r -'- 1)-th linear section onc morc point

According to cqu. (1.20) we obtain two cquations for dctcrmining th., t'I-O unknown yalues (Dr and J(r--l)'

4- Periodica Pulytl't:imic:l El. X,:!.

134

A characteristic feature of this method is that the length of the indi- yidual sections can be chosen arbitrarily. This means that in quantizing the describing function the yalue are taken in arbitrary points. The selection of points F{ and

F;C_l

plays a similar role. From the selection of these points depends the yalue of the obtained parameters which ensure in these points a giyen yalue of the describing function.

In reality in the case of an arbitrary selection

f(O)

and

Dr

(r 1 .. . m), a given value of the describing function in any point can be ensured by an appropriate selection of the slope of the linear sections. (Of course, the changes in the nonlinear characteristics may be yery sharp in this case.) For deter- mining these finite discontinuities a method has been proposed for which it was necessary that the describing function should haye giyen yalues also in other points. The values of these finite discontinuities can also be determined according to the formulae [6]

frO)

--- F,. b( Fr) ^:T 4

where

F,.

a yery small amplitude for which the yalue of the d('scrihing function is still given.

Dr

=

[b(cFr) - b(Fr)]

I (c) (2.8)

where (c 1) is a certain small number, particularly such a numher that for the amplitud(' B =

cFr

we are still at the ste('p section of thc function b(B).

eltimately, taking the established yalues

f(O)

and

Di

into consideration, th(' following expression is obtained according to formula (1.20) for deter- mining the inclination of the /z-th !"ection if the amplitude is B = F_{h :}

K, (2.9)

2. Tu:o valued i71L'erse characteristics

If

a

=

a(

B) 0, the inyerse nonlinear characteristic is two yalued.

The piecewise linear inverse characteristic can be determined hy means of the formulae (1.16), (1.19) and (1.22), (1.23).

DETERJILY.·JTIO,,' OF THE SO.YLLYEARITY 135 The shape of the nonlinear characteristic depends on the shape of the functions which determine the coefficients of the describing function.

a)

Let us assume that if B = 0, then

a(O)

and b(O) will be finite values (up to the point F_{l ).} From the equs. (1.16), (1.19) and (1.22), (1.23) it can be seen that in this case

]; (0) = f~ (0) = 0

From the same formulae ,\-e obtain:

b(O) =

~ ^(Ki

^l)

2

(2.10)

(2.11)

(the index in superscript brackets denotes the branch of the characteristics to which the parameters relate). From the formula (2.11) we obtain:

1:-0) _ 2b(O)

+

^;w(O)

A1 . - - - ' - - - ' -

:2 y(~) _ 2b(0)

:-z:a(O)

"-1 -

---2 ---

(2.12)

Let us assume that a( b) and b( B) are such functions that the inverse nonlinear characteristic has no finite discontinuities.

Applying the same procedure as for the single valued characteristics 'we sub-divide the horizontal axes of the functions a(b) and b(B) by the same selected points, and -we study one point after the other.

For the amplitude B = F~ we obtain according to formulae (1.16) and (1.22):

and according to formulae (1.19) and (1.23) we obtain:

-~~ ^IJ

(2.14)

Since

Ki

^l) and

Kf)

are kno,m from equ. (2.12), K~l) and K(2) can be deter- mined from equs. (2.13) and (2.14).

4*

136 J. so.uuJ

Generally, for the amplitude B =

Fh

we obtain:

h-l _~ '" (K(I) -_/ K\I) _/-:-1, ^...L

K(~)

_/ ^-

K(~)

₁₇₁ ^{) k}

(F t-'

_F ^{i_' )}

'1 _I

l=l , i,_

(2.15)

//-1

-:5'

(KO)

...

^{' /}

;=1

K\I,) _ K(2) --'-_/--1 1 i

K\~,»)

/ - ' - 1 -

I')

^F

Fi! _F~

FT 11

I! ._

(2.16)

From preyious calculation steps thc values

of the equs.

(2.15)

and

(2.16)

are kno'wn, and therefore from thcse

Kfzl)

and

Kh

²⁾ can he determined.

b)

The influence of finite discontinuities of nonlincar characteristics reflect strongly on the coefficients of the describing function in the same manner as for single yalued characteristics.

If

fl

(0) 0 and f~ (0) 0 (or only one of these), then, as can he es- tahlished from equs.

(1.16), (1.19)

and

(1.22), (1.23),

either the function a(B) or the function b( B) or both these functions will have an infinite discontinuity for the amplitude

B = O.

A.

finite discontinuity of the nonlinear characteristic produces an in- finite discontinuity in thc first derivatiyc of the function b(B) (if D~I) -D~:::»).

It is elear from Appendix 2 that a finite discontinuity in the charactcristic produces a finite discontinuity in the first deriyatin of the function a(B) (if D?) D~:::»).

Therefore, from the shape of the functions a( B) and b( B) it is easy to detect the presence of finitc discontinuitics in thc inyerse characteristics. Simi- larly, as in the case of single yalued characteristics

fl

(0),

f2

(0) and D~I), D~:::) can he determined according to the formulae

(1.16), (1.19),

and

(1.22), (1.23).

Selecting one more point inside thc giycn lineal' ,,('etion ,,-e ohtain further two equations 'whieh are necessary for determining the uukno'Hl parameters.

It 'was pointed out that for determining single valued i11yerSe charac- teristics it is possible to select arbitrarily the yaluc5 of the finite diseontinuities.

This fact remains correet eyell in the case of t'\\"o yalued inycrse characteristics.

Thm, the following yalues can bt' assumed [6]:

f1

(0) =

:-cF,. [b(F,.) --

a (Fr)]

.:1

f~ (0)

-1

[b(Fr)

a(F,.)]

(:"2.1 i)

DETEI'ULVATIO ... · OF THE .'·O.YLCiEARITY 137

In other locations the finite discontinuitics can he determined from the follow- ing equations:

D(l)

r 2

I dB ^{da '}

Fe..

^--

^ad

^{Ba .. '}

Fe-.; ¹

(2.18)

(2.19) Ultimately taking finite discontinuities into consideration, we ohtain the follo·wing equations for determining K,\l) and

Kj/)

from the equs. (1.16),

1,0 - - - - b

08 05

Q~ ~

02

100 150 200 B

- 06

- 03 1---",.---1-

;f. a

1,0 I---=---·~~---

Fig . .5

(1.19) and (1.22), (1.23):

and

b(FrJ

=2Jfl(0) - .DJO)]

:cF;,

1

J 2c'D

⁽⁰⁾

12

(0)] -'_ Ki,l)

:::T

l Fh

') D)1l -

Dj~)

Pi Fi

_F~

1J1. _J

h ,

(2.20)

h-l [

2

^(K~l)-

1=1

(2.21)

138 J. SOJIL6

This technique of determining inyerse nonlinear characteristics is fully applicahle eyen in the case of characteristics, the shape of 'which are frequency dependent. If the shape of the nonlinear characteristic depends on the amplitude of the input signal, then for a single specific amplitude we ohtain from the generalised formulae two equations at the most for determining a much larger numher of unknowns. Therefore, there is no single solution of the problem.

By

means of the descrihed method of determining the inyerse nonlinear characteristic the nonlinearity which descrihes the function illustrated in Fig. 3

50 K=O

Y 'to

JO _'t2

20 10

50 100 150 x

Fig. 6

is found in a form shown in Fig. 4. A detailed solution of this example is de- scrihed in an earlier work

[6],

,\-here al"o the example of determination of a two valued inyerse characteristic (Fig. 6) from known coefficient:;: of the describing function (Fig. ;)) is soh-cel.

Conclusions

A generalised method for determining the coefficients of thc describing function outlined at the beginning of this paper permits clirf'ct recording of thf' descrihing fUlletion of any pieeewise linear function with finite dis- continuities by means of formal steps. The thus obtained formulae haye a simple structure due to the presence in thcse of special functions. The numerical yalues of the coefficicnts of the describing fUIlctions can be easily determined by means of tahlcs.

The generalised method for determining the coefficients of the descrihing function will provide a direct possibility for determining the nonlinearity from its describing function. The thus determined piecewise linear nonlinear characteristic ,\-ith finite cliscontinuities will haye exact given yalues of the describing function only in selected points. This method is applicable for single yalued as well as for two yaluetl inycrse characteristics. The method is relatively simple to apply and suitable for soh-iag engineering prohlems.

DETEIDIISATIOS OF THE SOSLISEARITY

Appendix I

In equ. (1.20) the continuity of the function b( B) at the point B = 0 corresponds to the condition

f(O)

=

O.

It can be seen from the same formula that since k(I) = 1 and l(I) = 0, for amplitudes B> 0, the describing func- tion of any piece"wise linear characteristic "with finite discontinuities will be a continuous function.

From formulae (1.17) and (1.18) we obtain:

and

dl (

:i ^I

⁴ _]<'JJ~L~)

dB

^:r

Ba j!B2 - Fr

As can be seen from these expressions lim--dk =

O.

dB .

B-+F;

J. dt

lm =

=

dB

B-+Fi

For the amplitucl,"s B

>

Fi the deriYatiYes dk

dB

^and

functions.

ell

dB

^are continuous On the basis of what has been said aboyc the presence of a finite dis- continuity at the origin of the eoordinatp

[f(O)]

produces infinitc discontinuity of the function b(B) for B 0 and finite discontinuities at othcr points (Di) cause an infinite discontinuity of the first cleriYatiYe of the fUllction b( B).

Appendix 2

The first deriYatiYe of the coefficient aj (B) can be determincd from for- mula (1.I9). Since

o

140

and

lim

J. SOJILU

el( :~ ^- :LI

dB B-+Fi

da j

therefore at the amplitudes B

>

0 the function

·will

he continuous dB

except for those points ,,·here the nonlinear characteristic has a finite discon- tinuity. In the indicated point;; the function -_. has a finite discontinuity. da;

dB

According to equ. (1.19) the magnitude of these cliscontinuities will be

D

^{(j) -}_{r - : r}

F- ~

_r ¹¹

ela

dB F_{c . ,}

Summary da dB _FT-a!

I

This paper giyes a review of the general method for determining the describing function in case of piece wise linear nonlinear characteristics having discontinuities of the first kind.

This method establishes direct possibility for thc solntion of the inverse problem. that is for determinin!r the nonlinearity on the basis of its describin!r function. The inycrse character- istic is sOl~ght for in piece~\'ise linear nonlillear form with discontinuities of the first kind.

This nonlinear characteristic corresponds to the gi'·cn yalues of the describing function in the preselected points.

References

1. GIBSO:". J. E .• DITADA. E. S.: On the ill\·erse describing fnnction problem. Proc. 11. IFAC Con!rr .. Basle. 1963.

2. SO}ILO. ).:' Describing functions of hysteresis loop type nonlinearities approximated by piecewise linear sections. Second .\utomation Colloquium. Budapest. 1963. (in Hun·

garian) .

3. SmILo • .T.: Generalised method of harmonic linearisation of llonlincarities. ETA:'I I:\..

Conference. Lake Bled. 1964 (in Rus:,ian) .

. 1. SmILo. J.: Harmonic analysis of piecewise linear nonlincarities. 1. 11. :'IIcres

cs

Automatika.

(19M) 12, 2-3, (1965) (in HUIlgar·an).

5. GIBSO:", J. E.: ::"Ionlinear Automatic Control. :'IIcGraw·Hill Book Co .• 1963.

6. SO}ILO. J.: Determination of nonlinearities from the describing fUllction (the inverse de- scribing function prohlem). COIllmunications of the Automation Research Institute.

1965. :'10. 6 (in Hungarian).

Janos SQ:ULO, Budapest

::\'1.,

Egry J6zsef u.

18-20.

Hungary